THE GEOMETRY OF THE WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS

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1 THE GEOMETRY OF THE WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS OLEG IVRII Abstract. In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle S r = {z : z = r}, r 1, with an invariant subset G D called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations. Contents 1. Introduction 2 2. Background in Analysis Blaschke Products Petals and Flowers Quasiconformal Deformations Incompleteness: Special Case Renewal Theory Multipliers of Simple Cycles Lower bounds for the Weil-Petersson metric Incompleteness: General Case Limiting Vector Fields Asymptotics of the Weil-Petersson metric 44 References 48 This work is essentially a revised version of the author s PhD thesis at Harvard University. While at University of Helsinki, the author was supported by the Academy of Finland, project no

2 2 OLEG IVRII 1. Introduction 1.1. Basic notation. We write D for the unit disk and S 1 for the unit circle. Let m denote the Lebesgue measure on S 1, normalized to have unit mass. Given two points z 1, z 2 D, we denote the hyperbolic distance between them by d D (z 1, z 2 = inf We use the convention that the hyperbolic metric on the unit disk is ρ(z dz = 2 dz γ ρ. 1 z 2, dz while the Kobayashi metric is. The hyperbolic geodesic connecting the two 1 z 2 points is denoted by [z 1, z 2 ]. For z C \ {0}, let ẑ := z/ z. Let B p/q (η D be the horoball of Euclidean diameter η/q 2 which rests on e(p/q := e 2πi(p/q and H p/q (η = B p/q (η be its boundary horocycle. To compare quantities, we use: A B means A < const B, A B means A/B 1, A B means C 1 B < A < C 2 B for some constants C 1, C 2 > 0, A ɛ B means A/B 1 ɛ The traditional Weil-Petersson metric. To set the stage, we recall the definition and basic properties of the Weil-Petersson metric on Teichmüller space. Let T g,n denote the Teichmüller space of marked Riemann surfaces of genus g with n punctures. For a Riemann surface X T g,n, consider the spaces Q(X of holomorphic quadratic differentials with q <, X M(X of measurable Beltrami coefficients satisfying µ <. There is a natural pairing between quadratic differentials and Beltrami coefficients given by integration µ, q = µq. One has natural identifications X T XT g,n = Q(X, TX T g,n = M(X/Q(X. We will discuss two natural metrics on Teichmüller space: the Teichmüller metric and the Weil-Petersson metric. On the cotangent space, the Teichmüller and Weil- Petersson norms are given by ˆ q T = X ˆ q, q 2 WP = X ρ 2 q 2. The Teichmüller and Weil-Petersson lengths of tangent vectors are defined by duality, i.e. µ T := sup q T =1 µq and µ X WP := sup q WP =1 X µq. From the definitions, it is clear that the Teichmüller and Weil-Petersson metrics are invariant under the mapping class group Mod g,n. Petersson metric is not complete. However, unlike the Teichmüller metric, the Weil- For the Teichmüller space of a punctured torus T 1,1 = H, the mapping class group is Mod 1,1 = SL(2, Z. Let us denote the Weil-Petersson metric on T1,1 by ω T (z dz.

3 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 3 To describe the metric completion of (T 1,1, ω T, we introduce a system of disjoint horoballs. Let B 1/0 (η denote the horoball {z : y 1/η} that rests on = 1/0 and B p/q (η denote the horoball of Euclidean diameter η/q 2 that rests on p/q. For a fixed η 0, p/q Q { } B p/q(η is an SL(2, Z-invariant collection of horoballs. When η = 1, the horoballs have disjoint interiors but many mutual tangencies. We denote the boundary horocycles by H p/q (η := B p/q (η and H 1/0 (η := B 1/0 (η. Consider H with the usual topology. Extend this topology to H = H Q { } by further requiring {B p/q (η} η 0 to be open sets for p/q Q { }. Let us also consider a family of incomplete SL(2, Z-invariant model metrics ρ α on the upper half-plane: for α > 0, let ρ α be the unique SL(2, Z-invariant metric which coincides with the hyperbolic metric dz /y on H\ p/q Q { } B p/q(1 and is equal to dz /y 1+α on B 1/0 (1. Lemma 1.1. For α > 0, the metric completion of (H, ρ α is homeomorphic to H. Sketch of proof. To see that the irrational points are infinitely far away in the ρ α metric, notice that the horoballs {B p/q (2} cover the upper half-plane, while by SL(2, Z- invariance, the distance between H p/q (2 and H p/q (3 is bounded below in the ρ α metric. Therefore, any path γ that tends to an irrational number must pass through infinitely many protective shells B p/q (3 \ B p/q (2. In fact, this argument shows that an incomplete path γ is trapped within some horoball B p/q (3, from which it follows that it must eventually enter arbitrarily small horoballs. By the form of ρ α in B p/q (1, it is easy to see that the completion attaches only one point to the cusp at p/q. Theorem 1.1 (Wolpert. The Weil-Petersson metric on T 1,1 is comparable to ρ 1/2, i.e. 1/C ω T /ρ 1/2 C for some C > 1. Corollary. The metric completion of (T 1,1, ω T is homeomorphic to H Main results. In this paper, we replace the study of Fuchsian groups with complex dynamical systems on the unit disk D = {z : z < 1}. Inspired by Sullivan s dictionary, we are interested in understanding the Weil-Petersson metric on the space { } f : D D is a proper degree 2 map / B 2 = conjugacy by Aut(D. (1.1 with an attracting fixed point The multiplier at the attracting fixed point a : f f (p gives a holomorphic isomorphism B 2 = D. By putting the attracting fixed point at the origin, we can parametrize B 2 by a D : z f a (z = z z + a 1 + az. (1.2

4 4 OLEG IVRII All degree 2 Blaschke products are quasisymmetrically conjugate to each other on the unit circle, and except for the special map z z 2, they are quasiconformally conjugate on the entire disk. For this reason, it is somewhat simpler to work with B 2 := B 2 \ {z z 2 }, the quasiconformal moduli space M(f of a rational map described in [MS]. Given a Blaschke product f B 2, an f-invariant Beltrami coefficient on the unit disk µ M(D f defines a tangent vector in T f B 2. Since an f-invariant Beltrami coefficient descends to a Beltrami coefficient on the quotient torus of the attracting fixed point, M(D f = M(Tf. According to [MS], µ defines a trivial deformation in B 2 if and only if it defines a trivial deformation of T f T 1,1. In other words, one has a natural identification of tangent spaces T f B 2 = T Tf T 1,1 which shows that T 1,1 is the universal cover of B 2. To make the parallels with Teichmüller theory more explicit, we state our results on the universal cover. For this purpose, we pullback the Weil-Petersson metric on B 2 by a(τ = e 2πiτ to obtain a metric on T 1,1 = H, which we also denote ωb. Conjecture A. The metric ω B on T 1,1 = H is comparable to ρ1/4 on {τ : Im τ < 1}. In particular, the metric completion of (T 1,1, ω B is homeomorphic to H. In this paper, we show that 1/4 is the correct exponent in the conjecture above. More precisely, we show that: Theorem 1.2. The Weil-Petersson metric ω B on T 1,1 = H satisfies: (a ω B Cρ 1/4. (b There exists C small > 0 such that on p/q Q B p/q(c small, ω B (1/Cρ 1/4. Corollary. The Weil-Petersson metric on B 2 is incomplete. In fact, the Weil- Petersson length of each line segment e(p/q [1/2, 1 is finite. Corollary. The space H naturally embeds into the completion of (T 1,1, ω B. Remark. Since the Weil-Petersson metric is a real-analytic metric on B 2, the cusp at infinity in the H -model is somewhat special: w B Ce 2π Im τ dτ, as Im τ. Along radial rays a e(p/q, we have a more precise estimate: Theorem 1.3. Given a rational number p/q Q, as τ = p/q + it p/q vertically, the ratio ω B /ρ 1/4 C q, where C q is a positive constant independent of p. Conjecture B. We conjecture that C q is a universal constant, independent of q.

5 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 5 In a forthcoming work [Ivr], we will show that the Weil-Petersson metric is asymptotically periodic if we approach a e(p/q along a horocycle. The proof combines ideas from the work of Epstein [E] on rescaling limits with parabolic implosion Properties of the Weil-Petersson metric. In this section, we give a definition of the Weil-Petersson metric on B 2 B 2 in the form most useful for our later work. In Section 1.7, we will give equivalent definitions which work on the entire space B 2. For example, the Weil-Petersson metric may be described as the second derivative of the Hausdorff dimension of one-parameter families of Julia sets. It is convenient to put the Beltrami coefficient on the exterior unit disk. For a Beltrami coefficient µ M(D, we let µ + denote the reflection of µ in the unit circle: { µ + 0 for z D, = (1.3 (1/z µ for z S 2 \ D. Suppose X T g,n is a Riemann surface and µ M(X is a Beltrami coefficient. If X = D/Γ, we can consider µ as a Γ-invariant Beltrami coefficient on the unit disk. Let v be a solution of v = µ +. Since the set of all solutions is of the form v +sl(2, C, the third derivative v uniquely depends on µ +. As v is an infinitesimal version of the Schwarzian derivative, it is naturally a quadratic differential. In [McM2], McMullen observed that µ 2 WP 4 Area(X, ρ 2 ˆ 1 = I[µ] = lim r 1 2π z =r v 2 µ (z + ρ(z 2 dθ. (1.4 Similarly, given a Blaschke product f B 2, we can solve the equation v = µ + for µ M(D f. As above, a solution v of the equation v = µ + is well-defined up to adding a holomorphic vector field in sl(2, C so that v is uniquely defined. Following [McM2], we define the Weil-Petersson metric µ 2 WP := I[µ] provided that the limit exists. In Section 7, we will use renewal theory to establish the existence of this limit for any µ M(D f, invariant under a degree 2 Blaschke product other than z z 2. µ Figure 1. The support of the Beltrami coefficient takes up half of the quotient torus.

6 6 OLEG IVRII 1.5. A glimpse of incompleteness. We now sketch the proof of the upper bound in Theorem 1.2. To establish the incompleteness of the Weil-Petersson metric, we consider half-optimal Beltrami coefficients µ λ χ G(fa which take up half of the quotient torus at the attracting fixed point, but are sparse near the unit circle. Figure 2. Gardens G(f a for the Blaschke products with a = 0.5 and 0.8. Figure 3. A blow-up of G(f 0.5 near the boundary. A circle {z : z = r} with r close to 1 meets G(f 0.5 in small density. The garden G(f a D is an invariant subset of the unit disk whose quotient A = G(f a /f a T a is a certain annulus that takes up half of the Euclidean area of the quotient torus. To give an upper bound for the Weil-Petersson metric, we estimate the length of the intersection of G(f a with S r := {z : z = r}. In general, one has the estimate ( ωb ρ D 2 C lim sup G(f a S r. (1.5 r 1 In order for this estimate to be efficient, we take A to be a collar neighbourhood of the shortest p/q-geodesic in the quotient torus T fa T 1,1. For the Blaschke product f a with parameter a = e 2πiτ, τ H p/q (η, we prove lim sup G(f a S r = O(η 1/2. (1.6 r 1 Combining (1.5 and (1.6, we see that ω B Cρ 1/4 on {τ : Im τ < 1} as desired.

7 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 7 Remark. The trick of truncating the support of the Beltrami coefficient can be found in the proof of [McM1, Corollary 1.3]. See also [B] A glimpse of the convergence ω B /ρ 1/4 C q. We now sketch the proof of Theorem 1.3. To understand the behaviour of the Weil-Petersson metric as a e(p/q radially, we study the convergence of Blaschke products to vector fields. For example, as a 1 along the real axis, we will see that even though the maps f a (z = z z+a 1+az tend pointwise to the identity, their long-term dynamics tends to the flow of the holomorphic vector field κ 1 = z z 1. For the radial approach z+1 z a e(p/q, the maps f a (z e(p/qz converge pointwise to a rotation, and therefore the q-th iterates tend to the identity. We are thus led to extract a limiting vector field κ q by considering limits of the high iterates of f q a. It turns out that the vector field κ q is a q-fold cover of the vector field κ 1. In particular, it is independent of p. Figure 4. The vector fields κ 1 and κ 3. From the convergence of Blaschke products to vector fields, it follows that the flowers that make up the gardens G(f a for a e(p/q have nearly the same shape, up to affine scaling. Intuitively, for the integral average (1.4 to exist, when we replace r = 1 δ by r = 1 δ/2 say, we expect to intersect twice as many flowers to replenish the integral, i.e. we expect the number of flowers to be inversely proportional in δ. This leads us to explore an orbit counting problem for Blaschke products. The decay rate of the Weil-Petersson metric is governed by the dependence of the flower count on the parameter variable a Notes and references. In this section, we describe the space of Blaschke products of higher degree and equivalent definitions of the Weil-Petersson metric.

8 8 OLEG IVRII Blaschke products of higher degree. More generally, we can consider the space B d of marked Blaschke products of degree d which have an attracting fixed point modulo conformal conjugacy. By moving the attracting fixed point to the origin as before, one can parametrize B d by {a 1, a 2,..., a d 1 } D : z f a (z = z d 1 i=1 z + a i 1 + a i z. (1.7 Let a := a 1 a 2 a d 1 = f a(0 denote the multiplier of the attracting fixed point. It is because the maps are marked that we can distinguish the conformal conjugacy classes of a = {a 1, a 2,..., a d 1 } and ζ a = {ζa 1, ζa 2,..., ζa d 1 }. See [McM3] for more on markings. Mating. It is a remarkable fact that given two Blaschke products f a, f b of the same degree, one can find a rational map f a,b (z the mating of f a, f b whose Julia set is a quasicircle J a,b which separates the Riemann sphere into two domains Ω, Ω + such that on one side f a,b (z is conformally conjugate to f a, and to f b on the other. The mating is unique up to conjugation by a Möbius transformation. One can prove the existence of a mating by quasiconformal surgery (see [Mil] for details. The mating B d B d Rat d varies holomorphically with parameters. A natural way to put a complex structure on B d is via the Bers embedding B d P d which takes a Blaschke product and mates it with z d to obtain a polynomial of degree d. Here, the space P d = C d 1 is considered modulo affine conjugacy. The image of the Bers embedding is the generalized main cardioid in P d. Question. For d 3, what is the completion of B d with respect to the Weil-Petersson metric? Are the additional points precisely the geometrically finite parameters on the boundary of the generalized main cardioid? What is the topology on B d? Remark. Wolpert showed that the metric completion of (T g,n, ω T is the augmented Teichmüller space T g,n, the action of the mapping class group Mod g,n extends isometrically to (T g,n, ω T and the quotient M g,n = T g,n / Mod g,n is the Deligne-Mumford compactification. See [Wol] for more information. Equivalent definitions of the Weil-Petersson metric. Suppose f B d and f t, t ( ɛ, ɛ is a smooth path with f 0 = f, representing a tangent vector in T f B d. Consider the vector field v(z := d dt t=0 H 0,t (z where H 0,t : D Ω (f 0,t is the conformal conjugacy between f 0 and f 0,t. If f is a Blaschke product other than z z d, one can define f t 2 WP by the integral average (1.4, while if f(z = zd, one can use a more complicated integral average described in [McM2].

9 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 9 Remark. The definition of the Weil-Petersson metric via mating is slightly more general than the one via quasiconformal conjugacy given earlier because quasiconformal deformations do not exhaust the entire tangent space T f B d at the special parameters f B d that have critical relations. In [McM2], McMullen showed that f t 2 WP = 3 4 Var( φ, m log φ dm = 3 4 d2 dt 2 H. dim J 0,t (1.8 t=0 = 3 16 d2 dt 2 H. dim(h t,t m (1.9 t=0 where J 0,t is the Julia set of f 0,t, H t,t : S 1 S 1 is the conjugacy between f 0 and f t on the unit circle, (H t,t m is the push-forward of the Lebesgue measure, φ t = log f 0,t(H 0,t (z, log φ dm is the Lyapunov exponent, 1 Var(h, m := lim n Sn h(x 2 dm denotes the asymptotic variance in n the context of dynamical systems. Remark. Since J 0,t is a Jordan curve, H. dim J 0,t 1, so d dt t=0 H. dim J 0,t = 0 and d 2 t=0 H. dim J dt 2 0,t 0. Similarly, since (H t,t m is a measure supported on the unit d circle, H. dim(h t,t m 1, dt t=0 H. dim(h t,t m = 0 and d2 t=0 H. dim(h dt 2 t,t m Relations with quasiconformal geometry. The characterizations (1.8 and (1.9 of the Weil-Petersson metric are reflected in the duality between quasiconformal expansion and quasisymmetric compression: Theorem 1.4 (Smirnov [S]. For a k-quasiconformal map f : S 2 S 2, H. dim f(s k 2. Remark. If the dilatation µ(z = f f the stronger estimate H. dim f(s k 2 where k = is supported on the exterior unit disk, one has 2 k. 1+ k 2 Theorem 1.5 (Prause, Smirnov [PrSm]. For a k-quasiconformal map f : S 2 S 2, symmetric with respect to the unit circle, one has H. dim f m 1 k 2. An application of Theorem 1.4 or Theorem 1.5 shows: Corollary. The Weil-Petersson metric on B 2 is bounded above by 3/32 ρ D.

10 10 OLEG IVRII Proof. For a map f a B 2, the Bers embedding β fa gives a holomorphic motion of the exterior unit disk H : B 2 (S 2 \ D C given by H(b, z := H b,a (z. Note that the motion H is centered at a since H(a, is the identity. By the λ-lemma (e.g. see [AIM, Theorem ], one can extend H to a holomorphic motion H of the Riemann sphere satisfying µ H(b, b a. Observe that as d 1 ab D(b, a 0, b a 1 d 1 ab 2 D(b, a. Since each map H(b, is conformal on S 2 \ D, by the remark following Theorem 1.4, we have f t 2 WP 1 3 f 4 8 t 2 ρ D as desired. Acknowledgements. I would like to express my deepest gratitude to Curtis T. McMullen for his time, energy and invaluable insights. Binder for many interesting conversations. I also want to thank Ilia 2. Background in Analysis In this section, we explain how to bound the integral (1.4 in terms of the density of the support of µ. We also discuss a version of Koebe s distortion theorem for maps that preserve the unit circle Teichmüller theory in the disk. For a Beltrami coefficient µ, let v(z = v µ (z be a solution of the equation v = µ. The following formula is well-known (e.g. see [IT, Theorem 4.37]: for z supp µ. v (zdz 2 = ( 6 ˆ µ(ζ π C (ζ z 4 dζ 2 dz 2 (2.1 Lemma 2.1. For a Beltrami coefficient µ and a Möbius transformation γ Aut(S 2, we have v γ µ(z = v µ (γz γ (z 2 whenever γz supp µ. In particular, if µ is supported on the exterior of the unit disk and γ Aut(D, then v µ ρ (γ(z 2 = v γ µ ρ (z 2, z D. (2.2 Proof. The first statement follows from a change of variables and the identity γ (z 1 γ (z 2 (γ(z 1 γ(z 2 2 = 1 (z 1 z 2 2, z 1 z 2 C, γ Aut(S 2, (2.3 while the second statement follows from the fact that γ ρ = ρ for all γ Aut(D. To obtain upper bounds for the Weil-Petersson metric, we will use the following estimate:

11 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 11 Theorem 2.1. Suppose µ is a Beltrami coefficient which is supported on the exterior of the unit disk and has µ 1. Then, ˆ 1 v lim sup 2 µ (z r 1 2π ρ(z 2 dθ µ 2 lim sup supp µ SR. (2.4 R 1 + 2π z =r Theorem 2.2. Suppose µ is a Beltrami coefficient which is supported on the exterior of the unit disk and has µ 1. Let µ := (1/z µ be its reflection in the unit circle. Then, (a (v /ρ 2 (z 3/2 µ for z D. (b If d D (z, supp µ R then (v /ρ 2 (z e R. (c v /ρ 2 is uniformly continuous in the hyperbolic metric. Proof. By the Möbius invariance of v µ /ρ 2, it suffices to prove these assertions at the origin. Clearly, v (0 6 ˆ ˆ 1 π ζ >1 ζ dr 4 dζ r = 6. 3 Hence v /ρ 2 ( This proves (a. For (b, recall that d D(0, z = log(1 z + O(1. Then, v (0 6 π ˆ 1+Ce R > ζ >1 1 ζ 4 dζ 2 e R. 1 For (c, it suffices to observe that the kernel is uniformly continuous at z = 0 (ζ z 4 for {ζ : ζ > 1}. Proof of Theorem 2.1. Let V µ (z := 6 µ(ζ π ζ >1 ζ z 4 dζ 2. The calculation from part (a of Theorem 2.2 shows that V µ /ρ 2 3/2 µ has the same L bound. Set ν(ζ := 1 2π µ(e iθ ζ dθ. From Fubini s theorem, it is clear that ˆ ˆ V µ /ρ 2 dθ = V ν /ρ 2 dθ, 0 < r < 1. z =r z =r Since lim sup ζ 1 + ν(ζ µ lim sup 1 R 1 + 2π supp µ SR, ˆ 1 lim sup V µ (z r 1 2π ρ(z 2 dθ 3 2 µ 1 lim sup supp µ SR. R 1 + 2π z =r Equation (2.4 follows by multiplying the L 1 and L bounds A distortion theorem. The classical version of Koebe s distortion theorem says that if h : B(0, 1 C is univalent, then h (z 1 z for z < 1/2. We will mostly use a version of Koebe s distortion theorem for maps which preserve the real line or the unit circle:

12 12 OLEG IVRII Theorem 2.3. Suppose h : B(0, 1 C is a univalent function which satisfies h(0 = 0, h (0 = 1 and takes real values on ( 1, 1. For t < 1/2, h is nearly an isometry in the hyperbolic metric on B(0, t H, i.e. h ( dz /y t ( dz /y. In particular, h distorts hyperbolic distance and area by a small amount: Corollary. For z 1, z 2 B(0, t H, d H (z 1, z 2 = d H (h(z 1, h(z 2 + O(t. Corollary. If B is a round ball contained in B(0, t H, then Area (B, dz 2 t Area (h(b, dz 2. y 2 Above, A t B denotes that A/B 1 t. For a set E B(0, t, we call a set of the form h(e a t-nearly-affine copy of E. Suppose µ is a Beltrami coefficient supported on the upper half-ball B(0, 1 H. It is easy to see that for z B(0, t H, (h µ(z µ(h(z t µ where h µ = µ(h(z h (z. In terms of quadratic differentials, we have: h (z Lemma 2.2. On the lower half-ball B(0, t H, v v h (h(z µ (z 2 φ 1(t µ, (2.5 µ ρ for some function φ 1 (t satisfying φ 1 (t 0 + as t 0 +. ρ 2 Proof. Given R, ɛ > 0, we can choose t > 0 sufficiently small to guarantee that h (ζ 1 < ɛ and (z ζ ɛ (h(z h(ζ for z B(0, t H and ζ B = {w : d H (z, w < R}. y 2 Together with Theorem 2.3, these facts imply (2.5 with µ replaced by µ χ h(b. However, by part (b of Theorem 2.2, the contributions of µ(1 χ h(b and (h µ(1 χ B to (v µ /ρ 2 (h(z and (v h µ /ρ2 (z respectively are exponentially small in R Applications to Blaschke products. For a Blaschke product f B d, let δ c := min c D (1 c where c ranges over the critical points of f that lie inside the unit disk. By the Schwarz lemma, the post-critical set of f : S 2 S 2 is contained in the union of B(0, 1 δ c and its reflection in the unit circle. If ζ S 1, the ball B(ζ, δ c is disjoint from the post-critical set, and therefore all possible inverse branches f n are well-defined univalent functions on B(ζ, δ c. For 0 < t < 1/2, let U t := {z : 1 t δ c z < 1}. For Blaschke products, we have the following analogue of Lemma 2.2:

13 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 13 Lemma 2.3. If µ is an invariant Beltrami coefficient supported on the exterior unit disk, and if the orbit z f(z f n (z is contained in some U t with t < 1/2 sufficiently small, then v µ ρ (f n (z f n (z 2 v µ (z 2 z2 ρ2 φ 2(t µ, (2.6 for some function φ 2 (t satisfying φ 2 (t 0 + as t Blaschke Products In this section, we give background information on Blaschke products. We discuss the quotient torus at the attracting fixed point and special repelling periodic orbits called simple cycles on the unit circle. In the next section, we will examine the interface between these two objects Attracting tori. The dynamics of forward orbits of a Blaschke product f a (z = z z + a 1 + az (3.1 is very simple: all points in the unit disk are attracted to the origin. In this paper, we mostly assume that the multiplier of the attracting fixed point a = f (0 0. In this case, the linearizing coordinate ϕ a (z := lim n a n fa n (z conjugates f a to multiplication by a, i.e. ϕ a : D C, ϕ a (f a (z = a ϕ a (z. (3.2 It is well-known that (3.2 determines ϕ a uniquely with the normalization ϕ a(0 = 1. Let Ω denote the unit disk with the grand orbits of the attracting fixed and critical point removed. From the existence of the linearizing coordinate, it is easy to see that the quotient ˆϕ a : Ω T a := Ω/(f a is a torus with one puncture. We denote the underlying closed torus by T a. We will also consider the intermediate covering map π a : C T a = C /( a defined implicitly by ˆϕ a = π a ϕ a. Higher degree. For a Blaschke product f a B d with a = f a(0 0, the quotient torus T a has at most (d 1 punctures but there could be less if there are critical relations. The reader may view the space B d B d consisting of Blaschke products for which T a T 1,d 1 as a natural generalization of B Multipliers of simple cycles. On the unit circle, a Blaschke product has many repelling periodic orbits or cycles. Since all Blaschke products of degree 2 are quasisymmetrically conjugate on the unit circle, we can label the periodic orbits of f B 2 by the corresponding periodic orbits of z z 2.

14 14 OLEG IVRII A cycle is simple if f preserves its cyclic ordering. In this case, we say that ξ 1, ξ 2,..., ξ q has rotation number p/q if f(ξ i = ξ i+p (mod q. (For simple cycles, we prefer to index the points {ξ i } S 1 in counter-clockwise order, rather than by their dynamical order. Examples of cycles of degree 2 Blaschke products: (1, 2/3 has rotation number 1/2, (1, 2, 4/7 has rotation number 1/3, (1, 2, 3, 4/5 is not simple. In degree 2, for every fraction p/q Q/Z, there is a unique simple cycle of rotation number p/q. We denote its multiplier by m p/q := (f q (ξ 1. Since Blaschke products preserve the unit circle, m p/q is a positive real number (greater than 1. It is sometimes more convenient to work with L p/q := log(f q (ξ 1 which is an analogue of the length of a closed geodesic of a hyperbolic Riemann surface. 4. Petals and Flowers In this section, we give an overview of petals, flowers and gardens. As suggested by the terminology, gardens are made of flowers, and flowers are made of petals. We first give a general definition of a garden, but then we specify to half-flower gardens which will be used throughout this work. In fact, for a Blaschke product f a B 2, we will construct infinitely many halfflower gardens G [γ] (f a one for every outgoing homotopy class of simple closed curves [γ] π 1 (T a,. However, in practice, we use the garden G(f a := G [γ] (f a associated to the shortest geodesic γ in the flat metric on the torus. For parameters a B p/q (C small, the shortest curve γ is uniquely defined and has rotation number p/q. It is precisely for this choice of half-flower garden that the estimate (1.6 holds. For example, to study radial degenerations with a 1, we consider gardens where flowers have only one petal (see Figure 2, while for other parameters, it is more natural to use gardens where the flowers have more petals (see Figure 5 below Curves on the quotient torus. Inside the first homotopy group π 1 (T a, = Z Z, there is a canonical generator α which is represented by counter-clockwise loops ˆϕ a ({z : z = ɛ} with ɛ > 0 sufficiently small. By a neutral curve, we mean a curve whose homotopy class in π 1 (T a, is an integral power of α. All non-neutral curves can be classified as either incoming or outgoing, depending on their orientation: a curve γ : R/Z T a is outgoing if some (and hence every lift γ i = πa 1 γ i in C

15 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 15 satisfies Figure 5. The gardens G 1/2 (f 0.6 and G 1/3 (f 0.66 e 2πi/3. γ i (t + 1 = (1/a q γ i (t for some q 1. In other words, γ is outgoing if γ i (t as t. A curve is incoming if the opposite holds, i.e. if instead γ i (t 0 as t. A complementary (outgoing generator β is only canonically defined up to an integer multiple of α. In terms of the basis {α, β}, we say that an outgoing curve homotopic to (q pα + pβ has rotation number p/q. If we don t specify the choice of β, then p/q is only well-defined modulo Lifting outgoing curves. Suppose γ is a simple closed outgoing curve in T a of rotation number p/q mod 1. It has q lifts to C under the projection π a : C T a, which we denote γ1, γ2,..., γq. The curves γi are spirals that join 0 to. Each individual spiral is invariant under multiplication by a q. We typically index the spirals so that multiplication by a sends γi to γi+p. Let γ i := ϕ 1 a (γi be (further lifts in the unit disk emanating from the attracting fixed point. Lemma 4.1. Suppose γ is a simple closed outgoing curve in T a of rotation number p/q. Then, γ i joins the attracting fixed point at the origin to a repelling periodic point ξ i S 1 of rotation number number p/q. Proof. Pick a point z 1 on γ i, and approximate γ i by the backwards orbit of f q : z 1 z 2 z n... By the Schwarz lemma, the backwards orbit is eventually contained in U 1/2 = {z : 1 δ c /2 z < 1}, i.e. z n U 1/2 for n N. Since the Blaschke product is asymptotically affine, the hyperbolic distance d D (z n, z n+1 between successive points is bounded as it cannot substantially grow for n N. The boundedness of the backward jumps forces the sequence {z n } to converge to a repelling periodic point ξ i on the unit circle. The same argument shows that the hyperbolic length of the arc of γ i from z n to z n+1 is bounded, and therefore γ i itself

16 16 OLEG IVRII must converge to ξ i. Since f( γ i = γ i+p, we have f(ξ i = ξ i+p. Furthermore, since the lifts γ i D are disjoint, the points {ξ i } are arranged in counter-clockwise order which means that the repelling periodic orbit ξ 1, ξ 2,..., ξ q has rotation number p/q Definitions of petals and flowers. An annulus A T a homotopy equivalent in T a to an outgoing geodesic of rotation number p/q has q lifts in the unit disk emanating from the origin. We call these lifts petals and denote them Pi A, with i = 1, 2,..., q. Each petal connects the attracting fixed point to a repelling periodic point. Naturally, the flower is defined as the union of the petals: F = q i=1 PA i. We refer to the attracting fixed point as the A-point of the flower and to the repelling periodic points as the R-points. By construction, flowers are forward-invariant regions. The garden is the totally-invariant region obtained by taking the union of all the repeated pre-images of the flower: G = n=0 fa n (F. We refer to the iterated pre-images of petals and flowers as pre-petals and pre-flowers respectively. In degree 2, a flower has two pre-images: itself and an immediate preflower which we denote F for convenience. Each pre-flower has two proper preimages. We define the A and R points of pre-flowers as the pre-images of the A and R points of the flower. We typically label a pre-petal by its R-point and a pre-flower by its A-point Half-flower gardens. We now construct the special gardens that will be used in this work. For this purpose, observe that an outgoing homotopy class [γ] π 1 (T a, determines a foliation of the quotient torus T a by parallel lines, which are closed geodesics in the flat metric on T a. Explicitly, we can first foliate the punctured plane C by the logarithmic spirals γ θ := {e t log aq e iθ : t [, }, 0 θ < 2π, and then quotient out by ( a. The branch of log a q is chosen so that π a (γ θ [γ]. Note that since each individual spiral is only invariant under ( a q, a single line on the quotient torus T a corresponds to q equally-spaced spirals in C. Therefore, T a is foliated by the parallel lines γ θ := π a (γθ with 0 θ < 2π/q. For a Blaschke product f a B 2, the quotient torus T a has one puncture. Let A 1 = T a \ γ θc be the complement of the singular line that passes through this puncture. For 0 < α 1, let A α A 1 be the middle round annulus with Area(A α / Area(A 1 = α. By the construction of Section 4.3, the annulus A 1 defines a system of petals P 1 i,

17 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 17 i = 1, 2,..., q, which we calls whole petals. Similarly, an α-petal Pi α is defined as a petal constructed using the annulus A α T a. By default, we take α = 1/2 and write P i = P 1/2 i. We define the half-flower F as the union of all the half-petals. Alternatively, one can describe whole petals and half-petals in terms of linearizing rays. A linearizing ray, or a linearizing spiral if a / (0, 1, is defined as the preimage γ θ := ϕ 1 a (γθ, 0 θ 2π emanating from the attracting fixed point. If a whole petal P 1 consists of linearizing rays with arguments in (θ 1, θ 2 = ( x y, x+y, 2 2 then the associated α-petal P α is the union of the linearizing rays with arguments in ( x αy, x+αy. 2 2 Convention. In the rest of the paper, we use this system of flowers. When working with a e(p/q, we let F = F p/q denote the flower constructed from a foliation of the quotient torus by p/q-curves, arising from the choice of log a q log 1 = 0. Higher degree. One can similarly define petals and flowers similarly for Blaschke products of degree d 3: Call a line γ θ T a regular if it is contained in T a and singular if it passes through a puncture. The singular lines partition T a into annuli, the lifts of which we call whole petals. The number of (p/q-cycles of whole petals is at most d 1, but there could be less if several critical points lie on a single line. 5. Quasiconformal Deformations In this section, we describe the Teichmüller metric on B 2 and define the halfoptimal Beltrami coefficients which are supported on the half-flower gardens from the previous section. We also discuss pinching deformations. For a Beltrami coefficient µ with µ < 1, let w µ be the quasiconformal map fixing 0, 1, whose dilatation is µ. Given a rational map f(z Rat d, an invariant Beltrami coefficient µ M(S 2 f defines a (possibly trivial tangent vector in T f Rat d represented by the path f t = w tµ f (w tµ 1, t ( ɛ, ɛ. If µ M(D, one can also consider the symmetrized version w µ which is the quasiconformal map that has dilatation µ on the unit disk and is symmetric with respect to inversion in the unit circle. For a Blaschke product f B d and a Beltrami coefficient µ M(D f, the symmetric deformation f t = w tµ f (w tµ 1, t ( ɛ, ɛ, defines a path in B d. Note that while we use symmetric deformations to move around the space B d, we use asymmetric deformations w tµ + f (w tµ + 1 to compute the Weil-Petersson metric as the definition of µ WP involves v(z = d dt t=0 w tµ +(z.

18 18 OLEG IVRII The formula for the variation of the multiplier of a fixed point of a rational map will play a fundamental role in this work: Lemma 5.1 (e.g. Theorem 8.3 of [IT]. Suppose f 0 (z is a rational map with a fixed point at p 0 which is either attracting or repelling, and µ M(S 2 f 0. Then, f t = w tµ f 0 (w tµ 1 has a fixed point at p t = w tµ (p 0 and d dt log f t(p t = ± 1 ˆ t=0 π µ(z dz 2 (5.1 z 2 where T p0 is the quotient torus at p 0. The sign is + in the repelling case and in the attracting case Teichmüller metric. As noted in the introduction, T 1,1 is the universal cover of B 2 since one has an identification of the tangent spaces T fa B 2 = T Ta T 1,1. The Teichmüller metric on B 2 makes this correspondence a local isometry. More precisely, for a Beltrami coefficient µ M(D fa representing a tangent vector in T fa B 2, T p0 µ T (B 2 := ( ˆϕ a µ T (T1,1. A well-known result of Royden says that the Teichmüller metric on T 1,1 is equal to the Kobayashi metric; therefore, the same is true for the Teichmüller metric on B 2 = D. da a log a 2. Explicitly, the Teichmüller metric on B 2 is Lemma 5.1 distinguishes a one-dimensional subspace of Beltrami coefficients in M(D fa, namely ones of the form µ λ = ϕ a(λ (w/w (dw/dw with λ C. We refer to these coefficients as optimal Beltrami coefficients. Here, optimal is short for multiplier-optimal which refers to the fact that µ λ maximizes the absolute value of (d/dt t=0 log a t out of all Beltrami coefficients with L -norm λ. For a tangent vector v T T a T 1,1, the Teichmüller coefficient µ v associated to v is the unique Beltrami coefficient of minimal L norm which represents v. It is well-known that Teichmüller coefficients have the form λ q/ q with q Q(T a, where Q(T a is the space of integrable holomorphic quadratic differentials on the punctured torus T a. In particular, µ v T = sup q T =1 T µq a = µv. Since the quotient torus T a associated to a degree 2 Blaschke product f a B 2 has one puncture, Q(T a is one-dimensional. If we represent T a = C /( a, then Q(T a is spanned by (π a (dw 2 /w 2. Thus, in degree 2, the notions of Teichmüller coefficients and optimal coefficients agree. Higher degree. For a Blaschke product f a B d of degree d 3, the quotient torus has d 1 2 punctures, and so Q(T a Q(T a. Therefore, optimal Beltrami coefficients represent only a complex 1-dimensional set of directions in T T a T 1,d 1. In particular,

19 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 19 to understand the Weil-Petersson metric on spaces of Blaschke products of higher degree, one would need to study other deformations. Given an optimal Beltrami coefficient µ λ and a half-flower garden G(f a, we define the half-optimal Beltrami coefficient as µ λ χ G. Lemma 5.2. The half-optimal Beltrami coefficient µ χ G is half as effective as the optimal Beltrami coefficient µ, i.e. the map f t (µ χ G := w tµ χ G is conformally conjugate to f t(µ := w tµ f 0 (w tµ 1 d D (0, t = 2d D (0, t. f 0 (w tµ χ G 1 where t is chosen so that 5.2. Pinching deformations. A closed torus X = X τ = C/ 1, τ, τ H, carries a natural flat metric which is unique up to scale. To study lengths of curves on X, we normalize the total area to be 1. Given a slope p/q Q { }, let γ p/q X denote the Euclidean geodesic obtained by projecting (τ p/q R down to X. We define the pinching deformation (with respect to γ p/q as the geodesic in T 1 = H which joins τ to p/q. We further define the pinching coefficient µ pinch M(X as the Teichmüller coefficient which represents the unit tangent vector in the direction of this geodesic. Intrinsically, the pinching deformation is the most efficient deformation that shrinks the Euclidean length of γ p/q. More precisely, X t is the marked Riemann surface with d T (X, X t = 1 2 Teichmüller distance in T 1. log t+1 t 1 for which L X t (γ is minimal, where d T is the One can also define pinching deformations for annuli: given an annulus A = A 0, the pinching deformation (A t t 0 is the deformation for which the modulus of A t grows as quickly as possible. For the annulus A r,r := {z : r < z < R}, the pinching deformation is given by the Beltrami coefficients t µ pinch = t (w/w (dw/dw, t [0, 1. (5.2 With these definitions, the operation of pinching a torus X with respect to a Euclidean geodesic γ is the same as pinching the annulus A = X \ γ. Indeed, the modulus of X τ \ γ p/q is just mod(x τ \ γ p/q = Area X τ L Xτ (γ p/q 2 = { Im τ, qτ p 2 if p/q, Im τ, if p/q =. (5.3 The above formula appears in [McM4, Section 5], although McMullen normalizes the area of X τ to be Im τ. The modulus of course is independent of the normalization.

20 20 OLEG IVRII 6. Incompleteness: Special Case In this section, we show that the Weil-Petersson metric on B 2 is incomplete as we take a 1 along the real axis. As noted in the introduction, to show the estimate ω B /ρ D (1 a 1/4 on (1/2, 1], it suffices to prove: Theorem 6.1. For a Blaschke product f a B 2 with a [1/2, 1, we have lim sup r 1 We will deduce Theorem 6.1 from: G(f a S r = O ( 1 a. (6.1 Theorem 6.2. For a Blaschke product f a B 2 with a [1/2, 1, (a Every pre-petal lies within a bounded hyperbolic distance of a geodesic segment. (b The hyperbolic distance between any two pre-petals exceeds d D (0, a O(1. One curious feature of hyperbolic geometry is that a horocycle connecting two points is exponentially longer than the geodesic. Indeed, if x + iy, x + iy H, then the hyperbolic length of the horocycle joining them is 2(x/y while the geodesic length is only π θ dt = 2 log(cot(θ/2 where cot θ = x/y. As cot θ 1/θ for θ small, this θ sin t is approximately 2 log(2 x/y. With this in mind, we argue as follows: Proof of Theorem 6.1. By part (a of Theorem 6.2, the hyperbolic length of the intersection of S r with any single pre-petal is O(1. By part (b of Theorem 6.2, whenever the circle S r intersects a pre-petal, an arc of hyperbolic length O ( 1 a is disjoint from the other pre-petals. Therefore, only the O ( 1 a -th part of S r can be covered by pre-petals Quasi-geodesic property. Lemma 6.1. For a [1/2, 1, the petal P(f a lies within a bounded hyperbolic neighbourhood of a geodesic ray. Proof. By symmetry, the linearizing ray γ 0 = ϕ 1 a ((0, is the line segment (0, 1 which happens to be a geodesic ray. We therefore need to show that the petal P(f a = ϕ 1 a ({Re z > 0} lies within a bounded hyperbolic neighbourhood of γ 0. Suppose z P(f a lies outside a small ball B(0, δ. Let F be the fundamental domain bounded by {ζ : ζ = δ} and its image under f a. Under iteration, z eventually lands in F, e.g. z 0 = fa N (z F, with lim n arg f n (z ( π/2, π/2. On the other hand, the limiting argument of the critical point lim n arg f n (c = π since the forward orbit of the critical point is contained in the segment ( 1, 0. Therefore,

21 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 21 we can pick a point x 0 γ 0 for which d Ω (z 0, x 0 = d T a (π a (z 0, π a (x 0 = O(1. Let x = f N (x 0 be the N-th pre-image of x 0 along γ 0. Clearly, d D (z, x d Ω (z, x = d T a (π a (z 0, π a (x 0 = O(1. (6.2 This completes the proof The structure lemma. To establish the quasi-geodesic property for pre-petals, we show the structure lemma which says that the pre-petals are nearly-affine copies of the immediate pre-petal, while f : P 1 P is approximately the involution about the critical point, i.e. f P 1 m 0 c ( z m c 0, where m 0 c = z+c and 1+cz m c 0 = z c. For a Blaschke product f, its critically-centered version is given by 1 cz f = m c 0 f m 0 c. Naturally, the petals and pre-petals of f are defined as the images of petals and pre-petals of f under m c 0. Lemma 6.2 (Structure lemma. For a [1/2, 1 on the real axis, (i The critically-centered petal P B ( 1, const 1 a. (ii The immediate pre-petal P 1 B ( 1, const (1 a. Proof. Part (i follows from Lemma 6.1 as m c 0 ( (0, 1 = ( c, 1. To pin down the size and location of the immediate pre-petal, we use the fact that for a degree 2 Blaschke product, c is the hyperbolic midpoint of [0, a]. This implies that in the critically-centered picture, the A-point of the petal is m c 0 (0 = c while the A- point of the immediate pre-petal is m c 0 ( a = c. Therefore, by Koebe s distortion theorem, P 1 B ( 1, const 1 a. Part (ii follows by applying m 0 c to the last statement. Figure 6. Half-petal families for the Blaschke products f 0.8 and f 0.8.

22 22 OLEG IVRII 6.3. Petal separation. We can now prove that the petals are far apart: Proof of part (b of Theorem 6.2. Since the petal P is contained in a bounded hyperbolic neighbourhood of (0, 1 and the immediate pre-petal P 1 is contained in a bounded hyperbolic neighbourhood of ( 1, a, it follows that d D (P, P 1 = d D (0, a O(1. By the Schwarz lemma, given two pre-petals P ζ1 and P ζ2 with f n 1 (ζ 1 = f n 2 (ζ 2 = 1 and n 1 n 2 (say n 1 > n 2, d D (P ζ1, P ζ2 d D (f (n1 1 (P ζ1, f (n1 1 (P ζ2 d D (P 1, P 1. To complete the proof, it suffices to show that pre-petals P ζ1 and P ζ2 are far apart in the case that they have a common parent, e.g. when f(ζ 1 = f(ζ 2 = ζ. We prove this using a topological argument. Observe that 1 and 1 separate the unit circle in two arcs, each of which is mapped to S 1 \ {1} by f a. Therefore, any path in the unit disk connecting P ζ1 and P ζ2 must intersect the line segment ( 1, 1 P1 1 P 1. 1 However, we already know that the distance between P ζi to either P 1 and P 1 is greater than d D (0, a O(1 which tells us that the hyperbolic ( 1 d 2 D(0, a O(1- neighbourhood of ( 1, 1 is disjoint from P ζ1 and P ζ2. This completes the proof. 7. Renewal Theory In this section, we show that for a Blaschke product other than z z d, the integral average (1.4 defining the Weil-Petersson metric converges. The proof is based on renewal theory, which is the study of the distribution of repeated pre-images of a point. In the context of hyperbolic dynamical systems, this has been developed by Lalley [La]. We apply his results to Blaschke products, thinking of them as maps from the unit circle to itself. Using an identity for the Green s function, we extend renewal theory to points inside the unit disk. Renewal theory will also be instrumental in giving bounds for the Weil-Petersson metric. For a point x on the unit circle, let n(x, R denote the number of repeated preimages y (i.e. f n (y = x for some n 0 for which log (f n (y R. Also consider the probability measure µ x,r on the unit circle which gives equal mass to each of the n(x, R pre-images. We show: Theorem 7.1. For a Blaschke product f B d other than z z d, n(x, R e R log f dm as R. (7.1

23 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 23 Furthermore, as R, the measures µ x,r tend weakly to the Lebesgue measure. For a point z D, let N (z, R be the number of repeated pre-images of z that lie in the ball centered at the origin of hyperbolic radius R. Theorem 7.2. Under the assumptions of Theorem 7.1, we have N (z, R 1 2 log 1 z e R log f dm as R. (7.2 As before, when R, the N (z, R pre-images become equidistributed on the unit circle with respect to the Lebesgue measure Green s function. Let G(z = log 1 z be the Green s function of the disk with a pole at the origin. It is uniquely characterized by three properties: (i G(z is harmonic on the punctured disk, (ii G(z tends to 0 as z 1, (iii G(z log 1 z is harmonic near 0. Lemma 7.1. For a Blaschke product f B d, we have G(w i = G(z, z D. (7.3 f(w i =z To prove Lemma 7.1, it suffices to check that f(w i =z G(w i also satisfies the three properties above. We leave the verification to the reader. From equation (7.3, it follows that the Lebesgue measure on the unit circle is invariant under f. Indeed, for a point x S 1, one can apply the lemma to z = rx and take r 1 to obtain f(y=x f(y 1 = 1. (Alternatively, one can apply to both sides of (7.3 to obtain z the somewhat stronger statement f(w f(w=z = 1. wf (w In fact, the Lebesgue measure is ergodic. The argument is quite simple (see [SS] or [Ha]; for the convenience of the reader, we reproduce it here: given an invariant set E S 1, form the harmonic extension u E (z of χ E. Since χ f 1 E = χ E f, u E is a harmonic function in the disk which is invariant under f. But 0 is an attracting fixed point, so u E must actually be constant, which forces E to have measure 0 or 1 as desired. From the ergodicity of Lebesgue measure, it follows that conjugacies of distinct Blaschke products are not absolutely continuous Weak mixing. For the exceptional Blaschke product z z d, the pre-images of a point x S 1 come in packets and so n(x, R is a step function. Explicitly, n(x, R = 1 + d + d d log R/ log d.

24 24 OLEG IVRII While n(x, R has exponential growth, due to the lack of mixing, some values of R are special. All other Blaschke products satisfy the required mixing property and Theorem 7.1 follows from [La, Theorem 1 and formula (2.5]. Sketch of proof of Theorem 7.1. In the language of thermodynamic formalism, we must check that the potential φ f (x = log f (x is non-lattice, i.e. that the cohomology equation φ ψ = γ γ f does not admit solutions (ψ, γ with ψ(x valued in a discrete subgroup of R and γ(x bounded. To the contrary, the existence of such a pair of functions would imply that the multiplier spectrum {log(f n (ξ : f n (ξ = ξ} is contained in a discrete subgroup of R. Following the proof of [PP, Proposition 5.2], we see that there exists a function w C α (Σ satisfying w(f(x = e iaφ f (x w(x, for some a R \ {0}. (7.4 Here, Σ = {0, 1,..., d 1} N is the shift space which codes the dynamics of f on the unit circle. However, if we work directly on the unit circle and repeat the proof of [PP, Proposition 4.2], we obtain a function w C α (S 1 satisfying (7.4. Since w(x is non-vanishing and has constant modulus, we can scale it by a constant if necessary so that w(x = 1. By comparing the topological degrees of both sides of (7.4, we see that the topological degree of w is 0. In particular, w admits a continuous branch of logarithm. If w(x = e iv(x then v f = a φ f + v + 2πk for some constant k Z. Therefore, φ f 2πk/a is cohomologous to a constant. This tells us that the Lebesgue measure m must also be the measure of maximal entropy. However, the measure of the maximum entropy is a topological invariant, thus if we have a conjugacy h between z d and f(z, then the measure of the maximal entropy is h m. However, we know that the conjugacies of distinct Blaschke products are not absolutely continuous, therefore, we must have f(z = z d Computation of entropy. Since the dimension of the unit circle is equal to 1, the entropy h(f, m of the Lebesgue measure coincides with the Lyapunov exponent log f (e iθ dθ. We may compute the latter quantity using Jensen s formula: 1 2π Lemma 7.2. If a = f a(0 0, the entropy of the Lebesgue measure for the Blaschke product f a (z with critical points {c i } and zeros {z i } is given by ˆ 1 log f 2π a(e iθ dθ = G(c i G(a = G(c i G(z i. (7.5 cp cp zeros

25 WEIL-PETERSSON METRIC IN COMPLEX DYNAMICS 25 In particular, for degree 2 Blaschke products, as a tends to the unit circle, the entropy h(f a, m 1 c 2(1 a Laminated area. For a measurable set E in the unit disk, let Ê denote its saturation under taking pre-images, i.e. Ê = {ζ : f n (ζ E for some n 0}. For a saturated set Ê, we define its laminated area as A(Ê = lim r 1 1 E S 2π r and say that E subtends the A(Ê-th part of the lamination. By Koebe s distortion theorem (see Section 2.2, we have the following useful estimate: Lemma 7.3. Suppose E is a subset of U t := {z : 1 t δ c z < 1} with t < 1/2. If E is is disjoint from all of its pre-images, then ˆ A(Ê 1 1 t 2π h(f a, m 1 z dz 2. (7.6 (The notation A ɛ B means that A/B 1 ɛ. Proof. By breaking up the set E into little pieces, we may assume that E B(x, t for some x S 1. We claim that E 1 1 z dz 2 1 t f n (E 1 z dz 2, uniformly in n 0. By Lemma 2.2, for each n-fold pre-image E y of E, with f n (y = x, we have ˆ ˆ 1 E y 1 z dz 2 t (f n (y 1 1 E 1 z dz 2. The claim follows in view of the the identity f n (y=x (f n (y 1 = 1 (recall that the Lebesgue measure is invariant. Therefore, we may assume that E U t t > 0 arbitrarily small, i.e. we can pretend that f 1 is essentially affine. E with By approximation, it suffices to consider the case when E = R is a rectangle of the form {z : 1 z ( δ, (1 + ɛ 1 δ, arg z ( θ 0, θ 0 + ɛ 2 δ } with ɛ 1, ɛ 2 small. For k large, the circle S 1 δ/k = {z : z = 1 δ/k} intersects ɛ 1 k/h pre-images of R. As the hyperbolic length of S 1 δ/k is 2πk/δ and each pre-image has horizontal hyperbolic length ɛ 2, the laminated area A( ˆR ɛ 1ɛ 2 as desired. 2πh δ Recall from [McM2] that a continuous function h : D C is almost-invariant if for any ɛ > 0, there exists r(ɛ < 1, so that for any orbit z f(z f n (z contained in {z : r z < 1}, we have h(z h(f n (z < ɛ. Theorem 7.3. Suppose f is a Blaschke product other than z z d, and h is an almost-invariant function. Then the limit lim 1 r 1 h(zdθ exists. 2π z =r